Lallement Functor is a Weak Right Multiadjoint

Abstract For a plural signature $$\Sigma $$ Σ and with regard to the category $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$ NPIAlg ( Σ ) s , of naturally preordered idempotent $$\Sigma $$ Σ -algebras and surjective homomorphisms, we define a contravariant functor $$\textrm{Lsys}_{\Sigma }$$ Lsys Σ from $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$ NPIAlg ( Σ ) s to $$\textsf {Cat}$$ Cat , the category of categories, that assigns to $${\textbf {I}}$$ I in $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$ NPIAlg ( Σ ) s the category $${\textbf {I}}$$ I - $$\textsf {LAlg}(\Sigma )$$ LAlg ( Σ ) , of $${\textbf {I}}$$ I -semi-inductive Lallement systems of $$\Sigma $$ Σ -algebras, and a covariant functor $$(\textsf {Alg}(\Sigma )\,{\downarrow _{\textsf {s}}}\, \cdot )$$ ( Alg ( Σ ) ↓ s · ) from $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$ NPIAlg ( Σ ) s to $$\textsf {Cat}$$ Cat , that assigns to $${\textbf {I}}$$ I in $$\textsf {NPIAlg}(\Sigma )_{\textsf {s}}$$ NPIAlg ( Σ ) s the category $$(\textsf {Alg}(\Sigma )\,{\downarrow _{\textsf {s}}}\, {\textbf {I}})$$ ( Alg ( Σ ) ↓ s I ) , of the coverings of $${\textbf {I}}$$ I , i.e., the ordered pairs $$({\textbf {A}},f)$$ ( A , f ) in which $${\textbf {A}}$$ A is a $$\Sigma $$ Σ -algebra and "Equation missing" a surjective homomorphism. Then, by means of the Grothendieck construction, we obtain the categories $$\int ^{\textsf {NPIAlg}(\Sigma )_{\textsf {s}}}\textrm{Lsys}_{\Sigma }$$ ∫ NPIAlg ( Σ ) s Lsys Σ and $$\int _{\textsf {NPIAlg}(\Sigma )_{\textsf {s}}}(\textsf {Alg}(\Sigma )\,{\downarrow _{\textsf {s}}}\, \cdot )$$ ∫ NPIAlg ( Σ ) s ( Alg ( Σ ) ↓ s · ) ; define a functor $$\mathfrak {L}_{\Sigma }$$ L Σ from the first category to the second, which we will refer to as the Lallement functor; and prove that it is a weak right multiadjoint. Finally, we state the relationship between the Płonka functor and the Lallement functor.